A084385 a(1) = 1; a(n+1) is the smallest number not occurring earlier and coprime to Sum_{j=1..n} a(j).

1, 2, 4, 3, 7, 5, 9, 6, 8, 11, 13, 10, 12, 15, 17, 14, 16, 19, 21, 18, 20, 23, 25, 22, 24, 27, 29, 26, 28, 31, 33, 30, 32, 35, 37, 34, 36, 39, 41, 38, 40, 43, 45, 42, 44, 47, 49, 46, 48, 51, 53, 50, 52, 55, 57, 54, 56, 59, 61, 58, 60, 63, 65, 62, 64, 67, 69, 66, 68, 71, 73, 70

Offset: 1

Author

Amarnath Murthy, May 29 2003

Keywords

• nonn

Comments

Rearrangement of the positive integers.

Any sequence defined in this manner (that is, a(1) is any positive integer and a(n+1) is the smallest integer not occurring earlier and coprime to Sum_{j=1..n} a(j)) is a rearrangement of all positive integers. This property is used by problem 4 of Chinese High School Mathematical Olympiad in 2018. - Shu Shang, Sep 29 2021

Examples

			1+2+4 = 7, 3 is the smallest number not occurring earlier and coprime to 7, hence a(4) = 3.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Partial sums are in A111244. Cf. A064413.

Cf. A261351 (inverse).

Programs

Haskell

import Data.List (delete)
a084385 n = a084385_list !! (n-1)
a084385_list = 1 : f [2..] 1 where
   f xs s = g xs where
     g (y:ys) = if gcd s y == 1 then y : f (delete y xs) (s + y) else g ys
-- Reinhard Zumkeller, Aug 15 2015

PARI

used(k,v)=b=0; j=1; while(b<1&&j<=length(v),if(v[j]==k,b=1,j++)); b
{print1(s=1,","); v=[s]; for(n=1,72,j=1; k=2; while(used(k,v)||gcd(k,s)>1,k++); v=concat(v,k); s=s+k; print1(k,","))}

PARI

{print1(1,",",2,",",4,",",3,",",7,",",5,",");for(n=7,73,m=n%4;d=(if(m==0,-2,if(m==1,-1,if(m==2,1,2)))); print1(n+d,","))}

Formula

For n > 6: a(n) = n-2 for n mod 4 = 0, a(n) = n-1 for n mod 4 = 1, a(n) = n+1 for n mod 4 = 2, a(n) = n+2 for n mod 4 = 3. - Klaus Brockhaus, Nov 30 2003

Extensions

Edited, corrected and extended by Klaus Brockhaus, May 29 2003